Splay geometry with weak anchoring

When the applied field is low, the liquid crystal director is homogeneously aligned along the x axis and the tilt angle is 0. When the applied field is increased above a threshold, the liquid crystal director begins to tilt. When the applied field is only slighdy above the threshold, the tilt angle is small. Approximately, we have cos 9 1 and sin 9 9. Equations (5.58) and (5.59) become 

The Euler-Lagrange equation of the minimization of the total bulk free energy is (Equation (5.22)) 

Note that this is the solution for an applied field slightly above the threshold. The boundary 

When the anchoring is infinitely strong, IT = oo and thus L = 0 the solution to Equation (5.66) is hl2 = nl2, which gives the field threshold Ec = n/h) jK /soAs. When the anchoring is 

When the applied field is low, is large and 1. At the field threshold, hJ2 is close to nl2, and thus we have the approximation 

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